The ancient greeks where geometericists not algebraists, so for them the existence of a line was equivalent to the existence of what algebraists think of as a n

The ancient greeks where geometericists not algebraists, so for them the existence of a line was equivalent to the existence of what algebraists think of as a number. A square with unit length had a diagonal, and hence sqrt(2) existed. However, despite the name "Pythagoras' Theorem", sqrt(2) was traumatic for Pythagoras because he wanted to be believe in a rational universe, that is, one in which everything (distance of planets, musical harmonies and many other things) could be expressed as a ratio of whole numbers (rational). And then, damit, the simplest possible example, a unit square, threw a monkey wrench into the works! I have read that Pythagoreans were forbidden from talking about numbers like sqrt(2) (or from eating green beans).